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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 495408, 9 pages
Classical Solvability of a Free Boundary Problem for an Incompressible Viscous Fluid with a Surface Density Equation
Department of Mathematics, Faculty of Engineering, Tamagawa University, 6-1-1 Tamagawa-Gakuen, Machida, Tokyo 194-8610, Japan
Received 16 June 2013; Accepted 8 August 2013
Academic Editor: Rodrigo Lopez Pouso
Copyright © 2013 Yoshiaki Kusaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate a mathematical model introduced by Shikhmurzaev to remove singularities that arise when classical hydrodynamic models are applied to certain physical phenomena. The model is described as a free boundary problem consisting of the Navier-Stokes equations and a surface mass balance equation. We prove the local-in-time solvability in Hölder spaces.
Let a time-dependent bounded domain with the outer boundary be filled with an incompressible viscous fluid, and let represent the interface. In , we assume that the flow is governed by the Navier-Stokes equations: where is the velocity, is the pressure, and , which is assumed to be a positive constant, is the kinematic viscosity.
On , we assume the following equations: Here and are the velocity and the density of surface layer, respectively. is the stress tensor, where is the velocity deformation tensor. is the twice mean curvature of at the point , which is negative if is convex in the neighborhood of . is the unit outward normal to at the point . is the projection operator onto the tangent plane at the point on . denotes the derivative along the trajectory of particle on . is the gradient restricted to the surface. , , , , , are positive constants; in particular, is the density of the bulk and is the characteristic time scale over which the surface density relaxes to its equilibrium value .
Finally, to complete the problem, we give the initial conditions:
It is known that singularities arise when the the classical hydrodynamic equations and modeling assumptions are applied to certain physical phenomena. For example, the application of the classical no-slip boundary condition to the spreading of a drop on a plate gives rise to a nonintegrable shear stress, and the application of the classical kinematic condition at the free boundary to the formation of a cusp on a free surface of a viscous fluid leads to an infinite energy dissipation in the fluid (e.g., refer to  and the references therein).
To remove the above mentioned singularities, we are required to modify classical boundary conditions by taking into account molecular interaction near interfaces. The molecule in the liquid region which is very close to another phases experiences an asymmetric force due to the presence of another materials. This gives rise to the variation in the density in the liquid region near to the adjacent phase, and the surface tension occurs as a result of this variation in density. The thin layer in the liquid region in which the above mentioned density variation occurs is called the surface layer.
Through [1–4], Shikhmurzaev developed a theory to remove the above mentioned singularities by introducing a surface layer which is treated as a separate phase. In this theory, the no-slip condition assumed in classical models for dynamic wetting processes is modified as the Navier-slip condition through thermodynamic considerations on the surface layer (refer to [2, 3]). The formation of a free surface cusp associated with fluid flow is also investigated in . In , the cusp formation is modeled as an interface disappearance process. In this model, an internal surface stretching from the cusp, which is referred to as “the surface-tension-relaxation tail”, is introduced. The above mentioned singularity associated with the modeling of cusp formation arises owing to the absence of viscous stress at the cusp with which the surface tension acting from the liquid surface is balanced. In this model, the surface tension at the cusp can be balanced by shear stresses acting on this tail.
The problem (1)–(6) is a model describing the behavior of an isolated liquid drop in which the interface is modeled as a surface layer based on Shikhmurzaev’s theory. The dynamics of the liquid in this layer are governed by (3) which represents conservation of mass. The right-hand side of (3) represents the source consisting of a flow of molecules from the bulk. Equations in (4) are conditions that minimize the rate of entropy production in the surface layer. Equation (5) represents a linearized state equation in the surface layer (refer to  for details). In Shikhmurzaev’s theory, the surface layer is modeled as a sharp interface as a result of a continuum approximation. Thus, in the above problem, the surface layer is described by the equations given in (3)–(5) defined on a geometric surface, and the behavior of the surface layer is related to (1) in the bulk through the boundary conditions given in (2).
In the present paper, we prove the local-in-time classical solvability of problem (1)–(6). As is mentioned above, this model is important as a basic model to describe the above mentioned physical phenomena; however, as far as the author knows, any rigorous proofs on its solvability have not been given. In the present paper, we consider the case where the mass exchange between the interface and the bulk does not occur. As is seen in Section 2, under such an assumption, we can reformulate our problem as a problem defined in a domain with a fixed known boundary by introducing Lagrangian coordinates, and in Section 3, we construct a unique solution of the reformulated problem in Hölder spaces with the aid of the method of successive approximations.
2. Reformulation of the Problem
In this section, we reformulate our problem in Lagrangian coordinates. By Lagrangian coordinates we mean the initial coordinates of the fluid particles. In the case where no exchange of molecules occurs between the surface and the bulk, (4)1 is reduced to . This relation indicates that the following kinematic condition at the interface is satisfied: the interface consists of the particles located at the interface at the initial time. This circumstance enables us to relate each point to its initial point by relation (10) given below.
Before reformulating our problem, we rewrite (3) as a nonlinear parabolic equation on with the time derivative , where denotes the derivative along the trajectory of particle on the interface with velocity . Noting the following relation (e.g., see ): where represents the derivative along the trajectory which is normal to the interface, (3) can be written as Then eliminating from the above equation with the aid of the relation (4)2, we obtain the following equation:
Now let us reformulate our problem. The Lagrangian and Eulerian coordinates are related by where is the velocity at time of the particle which was located at at . By changing the variables from to by relation (10), problem (1)–(6) is reformulated as the following problem defined in the cylindrical domain with the lateral boundary : In (11)–(13), , and are , and , respectively. Consider ; here denotes the Jacobian matrix of , and the notation means the transpose of the matrix . is the outward unit normal to at the point , , and and are the operators defined by and , respectively. is the tensor with the elements , where is the -element of , and is Kronecker’s delta. The operators and are defined by where , , denote the components of the inverse matrix of , , denotes the local coordinates on , and denotes the components of the vector with respect to the basis , . Finally, the operator is defined as
Note that in derivation of (12)2, we have used the formula . Note also that although (12)1,2 are different from the following formulas which are obtained directly from (2): problem (11)–(13) is equivalent to problem (1)–(6) as far as the condition , which is valid for sufficiently small , is satisfied.
We now introduce some function spaces. Let be a domain in , let be a positive constant, let be a cylindrical domain , let be a nonnegative integer, and let , .
By , we define the space of functions , , with the norm for a multi-index .
By we denote an anisotropic Hölder space of functions whose norm is defined by where Here,
Finally, we introduce the function space equipped with the norm where
Now, let us state our main result.
Theorem 1. Let , be constants satisfying , . Assume that
Assume that there exist positive constants and such that and on . In addition, assume that the following compatibility conditions are satisfied:
where is the operator corresponding to with ; namely, is given by the formula in (14) with .
Then, for a positive constant , problem (11)–(13) has a unique solution with the following smoothness:
3. Proof of the Main Result
In this section, we will prove Theorem 1.
We begin with preparing estimates of solutions to some linear problems. For the following problem: the following result is given in . In (26), , , , are defined for a given vector in the same manner as , , , are defined, and is defined by (14) with .
Theorem 2. Let , and let , be positive constants satisfying , . Assume that
Assume that there exists a positive constant such that on . Assume that the following compatibility conditions are satisfied:
Assume that there exist functions , , , with a finite norm satisfying the relation
in the sense of distribution. Furthermore, assume that satisfies the inequality
for a sufficiently small positive constant .
Then problem (26) has a unique solution satisfying the following inequality: where and is a nondecreasing function of .
For the following problem: we have the following theorem. The assertion of the theorem immediately follows from the Hölder estimates for linear parabolic equations (e.g., see ).
Theorem 3. Let , and let be a positive constant satisfying . Assume that
Assume that there exists a positive constant such that on . Further assume the same assumptions for stated in Theorem 2.
Then, problem (32) has a unique solution satisfying the following inequality: where is a nondecreasing function of .
Theorem 4. Under the same assumptions given in Theorem 2 where only compatibility condition is replaced by and Theorem 3, problem (35) has a unique solution satisfying the following inequality: where is a nondecreasing function of .
In addition, we prepare estimates for , which are used later.
Lemma 5. Let and be the Jacobian matrices of the mappings and , respectively. Let us assume that and satisfy condition (30) for sufficiently small . Then, the following inequalities hold: where is a positive constant independent of , and for arbitrary , where is a positive constant depending only on .
Proof. In the following proof, , , and are positive constants independent of and . Let be the -component of .
Then, we have where denotes Kronecker’s delta. Then, using the inequality from (40), we have This inequality implies that holds for sufficiently small .
Now, let be the -components of , and let and be the cofactors of and , respectively.
Then from the inequalities we have
On the other hand, from the inequality which are obtained with the aid of the following inequality which holds for arbitrary : we have where is a positive constant depending only on . From (44) and (47), estimates (38) and (39) immediately follow. Thus, the proof is completed.
Now, let us prove Theorem 1 by the method of successive approximations. We take , and, for the known th approximation, we define the ()th approximation by the solutions of the following problem: where In the above formulas, is the operator corresponding to with , and denotes the operator obtained by differentiating the coefficients of with respect to .
Now, let us verify that all terms of the sequence are defined on some time interval independent of . We begin with the following lemma.
Lemma 6. Let be a constant satisfying . Then there exist positive constants and such that if and satisfy the following conditions: then the following inequality holds for a positive constant independent of , and :
Proof. In the proof, are positive constants independent of , , and .
Choosing as in the following interpolation inequality: we have the following estimate: With the aid of the above estimate, we can easily obtain the desired estimate for .
is estimated as follows. With the aid of the inequality we have On the other hand, with the aid of we have From (55) and (57), we have the desired estimate for . and are estimated in a similar manner. Thus, we have proved the lemma.
Now, let us take satisfying the following conditions: Here is assumed to be chosen, so that and . Since , the zeroth approximation obviously satisfies conditions (50) for the above . Hence, from (58), satisfies From (59), this inequality indicates that satisfies conditions (50), and hence we can obtain the same estimate as (60) for . Thus, repeating this argument, we can construct a sequence such that each term is defined on .
Let us proceed to the proof of the convergence of the sequence . In the following argument, denote positive constants independent of and represents various positive constants depending only on .
Let us set Subtracting (48) with index from that with index , we have where
Now, noting that the relations hold for the cofactors of the Jacobian matrix of any transformation from to , by direct calculations, we can verify that the following relations hold: where with In (66), the Einstein summation convention is used, denotes the -component of , and , , denote fundamental unit vectors in .
For the terms in (63), , and , we have the following estimate for arbitrary :
We will derive here the estimate only of because the other terms can be estimated in a similar and simpler manner. Noting that the following estimates hold for and : and with the aid of estimates (38) and (39), we have
Noting also the following inequality: from estimate (69) for , we have
Now, applying Theorem 4 to problem (62), we have the following estimate: where the norm is defined as . Fixing so that and summing the above inequalities from to , we have where Then, using Gronwall’s inequality, from the above inequality, we have
Noting that the right-hand side in (75) is independent of , we can conclude that the sequence is convergent in .
Now, let us prove Theorem 1. Taking the limit as tends to infinity in problem (48), we can easily see that the limit of the sequence is a solution of problem (11)–(13). The uniqueness can be proved as follows. Let and be two solutions of problem (11)–(13). By subtracting one equation from the other, we obtain the equations for the differences , , and , the form of which is similar to (62). Then, in a similar manner to obtain (72), we can obtain the following estimate: This inequality implies that and , and as a consequence, follows. Thus, the proof of Theorem 1 is completed.
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